1.2 Model systems

Statistical physics must make use of models. To simplify matters we sometimes invoke crudely simplified images of reality, which may seem to render the results of theory useless. However, many fundamental statements are all but independent of the particular model system used in their derivation. Instead, they are consequences of a few basic properties common to all many-body systems, and of the laws of statistics. Thus the following models should be understood primarily as vehicles for demonstrating statistical-mechanical predictions.

• STADIUM BILLIARD
  This model, introduced by Bunimovich and Sinai, will serve us to demonstrate the existence of chaos even in quite small systems. Chaos, it must be understood, is a fundamental prerequisite for the application of statistical rules. It is a basic property of chaotic systems that they will acquire each one of a certain set of ``states'' with equal probability, i.e. with equal relative frequency. It is this ``equal a priory probability'' of states which we need to proceed into the heart of Statistical Mechanics.

  The stadium billiard is defined as follows. Let a ``light ray'' or ``mass point'' move about in a two-dimensional container with reflecting or ideally elastic walls. The boundaries have both flat and semicircular parts, which gives rise to an efficient mixing of the flight directions upon reflection. Such a system is chaotic, meaning that the motional degrees of freedom $\{v_{x},v_{y}\}$ exhibit a uniform probability distribution. Since we have $v_{x}^{2}+v_{y}^{2}=const$ the ``phase points'' describing the momentary motion are distributed evenly over the periphery of a circle. The model is easily extended into three dimensions; the velocity then has three components, energy conservation keeps the phase points on the surface of a sphere, and equal a priory probability (or chaos) means that the distribution of points on this ``energy surface'' is homogeneous - nowhere denser or thinner than elsewhere.
  
  

  

  
  Applet Stadium: Start

  Simulation: Stadium Billiard in 2 Dimensions
  - See the trajectory of the particle (ray); note the frequency histograms for flight direction $\phi \equiv arctan(v_{y}/v_{x})$ and $x$-velocity $v_{x}$.
  - ``Chaos'' is demonstrated by simultaneously starting a large number of trajectories with nearly identical initial directions: $\rightarrow$ fast emergence of equidistribution on the circle $v_{x}^{2}+v_{y}^{2}=const$.
  [Code: Stadium]
  
  

  

  Applet VarSinai: Start

  
  Simulation: Sinai Billiard
  - Ideal one particle gas in a box having randomizers on its walls
  - See the trajectory of the particle (ray); note the frequency histograms for flight direction $\phi \equiv arctan(v_{y}/v_{x})$ and $x$-velocity $v_{x}$
  [Code: VarSinai]
  
  

• CLASSICAL IDEAL GAS
  $N$ particles are confined to a volume $V$. There are no mutual interactions between molecules - except that they may exchange energy and momentum in some unspecified but conservative manner. The theoretical treatment of such a system is particularly simple; nevertheless the results are applicable, with some caution, to gases at low densities. Note that air at normal conditions may be regarded an almost ideal gas.

  The momentary state of a classical system of $N$ point particles is completely determined by the specification of all positions $\{ \vec{r}_{i} \}$ and velocities $\{ \vec{v}_{i} \}$. The energy contained in the system is entirely kinetic, and in an isolated system with ideally elastic walls remains constant.
  
  

• IDEAL QUANTUM GAS
  Again, $N$ particles are enclosed in a volume $V$. However, the various states of the system are now to be specified not by the positions and velocities but according to the rules of quantum mechanics. Considering first a single particle in a one-dimensional box of length $L$. The solutions of Schroedinger's equation
  

  $${\cal H} \Psi(x) = E \Psi(x) \;\; with \; {\cal H} \equiv - \frac{\hbar^{2}}{2m} \nabla^{2}$$ (1.19)

  
  
  are in this case $\Psi_{n}(x) = (1/\sqrt{L}) \sin(2 \pi n x/L)$, with the energy eigenvalues $E_{n}= h^{2}n^{2}/8mL^{2}$ ($n=1,2,\dots$).
  
  

  Figure 1.4: Quantum particle in a 2-dimensional box: $n_{x}=3$, $n_{y}=2$

  

  In two dimensions - the box being quadratic with side length $L$ - we have for the energies
  

  $$E_{\vec{n}}= \frac{h^{2}}{8mL^{2}}(n_{x}^{2}+n_{y}^{2})$$ (1.20)

  
  
  where $\vec{n} \equiv \{ n_{x}, n_{y} \}$. A similar expression may be found for three dimensions.

  If there are $N$ particles in the box we have, in three dimensions,
  

  $$E_{N,\vec{n}}= \frac{h^{2}}{8mL^{2}} \sum_{i=1}^{N} [n_{ix}^{2}+n_{iy}^{2}+n_{iz}^{2}]$$ (1.21)

  
  
  where $n_{ix}$ is the $x$ quantum number of particle $i$.

  Note: In writing the sum 1.21 we only assumed that each of the $N$ particles is in a certain state $\vec{n}_{i}$ $\equiv (n_{ix},n_{iy},n_{iz})$. We have not considered yet if any combination of single particle states $\vec{n}_{i} \; (i=1, \dots, N)$ is indeed permitted, or if certain $\vec{n}_{i}, \vec{n}_{j}$ might exclude each other (Pauli priciple for fermions.)
  
  

• HARD SPHERES, HARD DISCS
  Again we assume that $N$ such particles are confined to a volume $V$. However, the finite-size objects may now collide with each other, and at each encounter will exchange energy and momentum according to the laws of elastic collisions. A particle bouncing back from a wall will only invert the respective velocity component.

  At very low densities such a model system will of course resemble a classical ideal gas. However, since there is now a - albeit simplified - mechanism for the transfer of momentum and energy the model is a suitable reference system for kinetic theory which is concerned with the transport of mechanical quantities. The relevant results will be applicable as first approximations also to moderately dense gases and fluids.

  In addition, the HS model has a special significance for the simulation of fluids.
  
  

  

  
  Applet Harddisks: Start

  $\parbox{360pt}{ \mbox{}\\ {\bf Simulation: Hard discs} \mbox{}\\ -... ...}/v_{x})$ and $x$-speed $v_{x}$. \\ \vspace{2ex} {\small [Code: Harddisks]} }$
  
  

  

  
  Applet Hspheres: Start

  $\parbox{360pt}{ \mbox{}\\ {\bf Simulation: Hard spheres} \mbox{} ... ...y}/v_{x})$ and $x$-speed $v_{x}$. \\ \vspace{2ex} {\small [Code: Hspheres]} }$
  
  $\Rightarrow \;$ Simulation 1.4: $N$ hard discs in a 2D box with periodic boundaries. [Code: Hdiskspbc, UNDER CONSTRUCTION]
  
  

• LENNARD-JONES MOLECULES
  This model fluid is defined by the interaction potential (see figure)
  

  $$U(r) = 4 \epsilon [(r/\sigma)^{-12} - (r/\sigma)^{-6}]$$ (1.22)

  
  
  where $\epsilon$ (potential well depth) and $\sigma$ (contact distance) are substance specific parameters.

  In place of the hard collisions we have now a continuous repulsive interaction at small distances; in addition there is a weak attractive force at intermediate pair distances. The model is fairly realistic; the interaction between two rare gas atoms is well approximated by equ. 1.22.
  
  

  Figure 1.7: Lennard-Jones potential with $\epsilon =1$ (well depth) and $\sigma = 1$ (contact distance where $U(r)=0$). Many real fluids consisting of atoms or almost isotropic molecules are well described by the LJ potential.

  
  The LJ model achieved great importance in the Sixties and Seventies, when the microscopic structure and dynamics of simple fluids was an all-important topic. The interplay of experiment, theory and simulation proved immensely fruitful for laying the foundations of modern liquid state physics.

  In simulation one often uses the so-called ``periodic boundary conditions'' instead of reflecting vessel walls: a particle leaving the (quadratic, cubic, ...) cell on the right is then fed in with identical velocity from the left boundary, etc. This guarantees that particle number, energy and total momentum are conserved, and that each particle is at all times surrounded by other particles instead of having a wall nearby. The situation of a molecule well within a macroscopic sample is better approximated in this way.
  
  

  

  
  Applet LJones: Start

  $\parbox{360pt}{ \mbox{}\\ {\bf Simulation: Lennard-Jones fluid} \mb... ...x with periodic boundary conditions. \\ \vspace{2ex} {\small [Code: LJones]} }$
  
  

• HARMONIC CRYSTAL
  The basic model for a solid is a regular configuration of atoms or ions that are bound to their nearest neighbors by a suitably modelled pair potential. Whatever the functional form of this potential, it may be approximated, for small excursions of any one particle from its equilibrium position, by a harmonic potential. Let $l$ denote the equilibrium (minimal potential) distance between two neighbouring particles; the equilibrium position of atom $j$ in a one-dimensional lattice is then given by $X_{j}=j l$. Defining the displacements of the atoms from their lattice points by $x_{j}$ we have for the energy of the lattice
  

  $$E(\vec{x},\dot{\vec{x}}) = \frac{f}{2} \sum_{j=2}^{N} \left(... ...{j-1} \right)^{2} + \frac{m}{2} \sum_{j=1}^{N} \dot{x}_{j}^{2}$$ (1.23)

  
  
  where $f$ is a force constant and $m$ is the atomic mass.

  The generalization of 1.23 to $2$ and $3$ dimensions is straightforward.

  The further treatment of this model is simplified by the approximate assumption that each particle is moving independently from all others in its own oscillator potential: $E(\vec{x},\dot{\vec{x}})$ $\approx f \sum_{j=1}^{N} x_{j}^{2}$ $+ (m/2) \sum_{j=1}^{N} \dot{x}_{j}^{2}$. In going to $2$ and $3$ dimensions one introduces the further simplifying assumption that this private oscillator potential is isotropically ``smeared out'': $E_{pot}(\{x,y,z\})$ $\approx f \sum_{j=1}^{N} (x_{j}^{2}+y_{j}^{2}+z_{j}^{2})$. The model thus defined is known as the ``Einstein model'' of solids.
  
  

• MODELS FOR COMPLEX MOLECULES
  Most real substances consist of more complex units than isotropic atoms or Lennard-Jones type particles. There may be several interaction centers per particle, containing electrical charges, dipoles or multipoles, and the units may be joined by rigid or flexible bonds. Some of these models are still amenable to a theoretical treatment, but more often than not the methods of numerical simulation - Monte Carlo or molecular dynamics - must be invoked.
  
  
  
  
• SPIN LATTICES
  Magnetically or electrically polarizable solids are often described by models in which ``spins'' with the discrete permitted values $\sigma_{i} = \pm 1$ are located at the vertices of a lattice.

  If the spins have no mutual interaction the discrete states of such a system are easy to enumerate. This is why such models are often used to demonstrate of statistical-mechanical - or actually, combinatorical - relations (Reif: Berkeley Lectures). The energy of the system is then defined by $E \equiv - H \sum_{i} \sigma_{i}$, where $H$ is an external field.

  Of more physical significance are those models in which the spins interact with each other, For example, the parallel alignment of two neighboring spins may be energetically favored over the antiparallel configuration (Ising model). Monte Carlo simulation experiments on systems of this type have contributed much to our understanding of the properties of ferromagnetic and -electric substances.